Question 8: What is meant by the dimensions of physical quantities? What are limitations and applications of dimensional analysis?
Dimensions of physical quantity
“It is the number of times the fundamental quantities (mass, length, time etc.) appear in a physical quantity.”
Dimensions relate the derived quantity to the base quantities. It enables us to know how a (derived) physical quantity is related to the base quantities of length, mass, time and other base quantities. Dimensions are represented by capital letters enclosed in squared brackets.
- Dimensions of base quantities
Each of the seven base quantities of SI have its own dimensions which is represented by a single sans-serif roman capital letter. These dimensions are given in the table below.
- Dimensions of derived quantities
Dimensions of derived quantities are the products of the dimensions of the base quantities from which they are derived. Let’s take some examples:
- Area of a square is obtained by multiplying the two of its sides. Therefore, the unit of area is m2. Replace the unit of length by the corresponding dimensions, we have dimensions of area = [L] [L] = [L2].
- Unit of acceleration is given by ms-2. Replace the base units (meter and second) by the corresponding dimensions, we have dimensions of acceleration = [L] [T-2] = [LT-2].
Generally, the dimensions of any quantity Q is written in the form of dimensions as follow;
Dimensions of some derived quantities are given in the following table.
Applications of dimensions
- Only quantities of the same dimensions can be mathematically operated (added, subtracted, multiplied, divided etc.) For example, we cannot add 2 kg of sugar with 10 seconds of time because both have different dimensions, therefore, in an equation both sides must have same dimensions. So dimensions can be used to check the correctness of an equation.
- Dimensional analysis can be used to derive the possible formula of a physical quantity.
- To derive equations showing the relation between different physical quantities.
Limitations of dimensions
Although dimensional analysis is very useful, it cannot help us more due to the following reasons:
- Dimensionless quantities cannot be determined by this method.
- Physical quantities represented by a large number of base quantities are difficult to handle with this method.
- If there are addition and subtraction appearing on one side of the equation, we cannot apply this method.
- This is not applicable to logarithmic, trigonometric or exponential functions.